| L(s) = 1 | + 2·4-s + 7-s + 5·13-s − 8·19-s − 10·25-s + 2·28-s − 14·31-s + 10·37-s + 13·43-s + 7·49-s + 10·52-s + 13·61-s − 8·64-s − 11·67-s + 34·73-s − 16·76-s − 26·79-s + 5·91-s − 5·97-s − 20·100-s − 26·103-s − 38·109-s + 11·121-s − 28·124-s + 127-s + 131-s − 8·133-s + ⋯ |
| L(s) = 1 | + 4-s + 0.377·7-s + 1.38·13-s − 1.83·19-s − 2·25-s + 0.377·28-s − 2.51·31-s + 1.64·37-s + 1.98·43-s + 49-s + 1.38·52-s + 1.66·61-s − 64-s − 1.34·67-s + 3.97·73-s − 1.83·76-s − 2.92·79-s + 0.524·91-s − 0.507·97-s − 2·100-s − 2.56·103-s − 3.63·109-s + 121-s − 2.51·124-s + 0.0887·127-s + 0.0873·131-s − 0.693·133-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 13689 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 13689 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.327336550\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.327336550\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 3 | | \( 1 \) |
| 13 | $C_2$ | \( 1 - 5 T + p T^{2} \) |
| good | 2 | $C_2^2$ | \( 1 - p T^{2} + p^{2} T^{4} \) |
| 5 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 7 | $C_2$ | \( ( 1 - 5 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 11 | $C_2^2$ | \( 1 - p T^{2} + p^{2} T^{4} \) |
| 17 | $C_2^2$ | \( 1 - p T^{2} + p^{2} T^{4} \) |
| 19 | $C_2$ | \( ( 1 + T + p T^{2} )( 1 + 7 T + p T^{2} ) \) |
| 23 | $C_2^2$ | \( 1 - p T^{2} + p^{2} T^{4} \) |
| 29 | $C_2^2$ | \( 1 - p T^{2} + p^{2} T^{4} \) |
| 31 | $C_2$ | \( ( 1 + 7 T + p T^{2} )^{2} \) |
| 37 | $C_2$ | \( ( 1 - 11 T + p T^{2} )( 1 + T + p T^{2} ) \) |
| 41 | $C_2^2$ | \( 1 - p T^{2} + p^{2} T^{4} \) |
| 43 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 - 5 T + p T^{2} ) \) |
| 47 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 53 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 59 | $C_2^2$ | \( 1 - p T^{2} + p^{2} T^{4} \) |
| 61 | $C_2$ | \( ( 1 - 14 T + p T^{2} )( 1 + T + p T^{2} ) \) |
| 67 | $C_2$ | \( ( 1 - 5 T + p T^{2} )( 1 + 16 T + p T^{2} ) \) |
| 71 | $C_2^2$ | \( 1 - p T^{2} + p^{2} T^{4} \) |
| 73 | $C_2$ | \( ( 1 - 17 T + p T^{2} )^{2} \) |
| 79 | $C_2$ | \( ( 1 + 13 T + p T^{2} )^{2} \) |
| 83 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 89 | $C_2^2$ | \( 1 - p T^{2} + p^{2} T^{4} \) |
| 97 | $C_2$ | \( ( 1 - 14 T + p T^{2} )( 1 + 19 T + p T^{2} ) \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{4} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−13.68878978936650234687996714425, −13.29495144815058327557878512138, −12.71655998394923824990111930580, −12.30038021614666650701412544902, −11.46977469887847279077121922683, −11.21867273264163268591275018335, −10.82091562245990221907594748138, −10.42730372558088741522092921022, −9.359902462929009005981090779922, −9.175929646603444156515207466466, −8.197187863482219290864543145817, −7.969176601547087292910961275462, −7.12479528520181278145061038849, −6.63122969102480163207734998762, −5.81473512122248334257704625242, −5.65501082476237562525325714497, −4.06267316216209923172480215955, −4.02105013198232063605481217469, −2.52895160986709551773881672407, −1.80387006944753394195323284491,
1.80387006944753394195323284491, 2.52895160986709551773881672407, 4.02105013198232063605481217469, 4.06267316216209923172480215955, 5.65501082476237562525325714497, 5.81473512122248334257704625242, 6.63122969102480163207734998762, 7.12479528520181278145061038849, 7.969176601547087292910961275462, 8.197187863482219290864543145817, 9.175929646603444156515207466466, 9.359902462929009005981090779922, 10.42730372558088741522092921022, 10.82091562245990221907594748138, 11.21867273264163268591275018335, 11.46977469887847279077121922683, 12.30038021614666650701412544902, 12.71655998394923824990111930580, 13.29495144815058327557878512138, 13.68878978936650234687996714425
